4 THE PRINCIPLE OF ANALYTIC CONTINUATION 209 (9.4.4) Let A c Cp be an open connected set, f and g two analytic functions in A with values in a complex Banach space E. Let U be an open subset of A, b a point of U, and suppose that f(x) ~g(x) in the set U n (b + Rp); thenf(x) = g(x)for every x e A. We can suppose, by a translation, that b = 0; let h=f—gy and let P be a polydisk in (7, of center 0, contained in U and such that in P, h(z) is equal to the sum of an absolutely summable power series (cvzv). Now P n Rp is a polydisk in Rp, and h(x) = 0 in P n W; this shows by (9.1.6) that cv = 0 for every v, hence h(z) = 0 in P and (9.4.2) can be applied. In an open connected set A c Kp, we say that a subset M c A is a set of uniqueness if any two functions, defined and analytic in A, coincide in A as soon as they coincide in M. (9.4.2), (9.4.3), and (9.4.4) show that a non- empty open subset U of A, or the intersection U n (b + Rp) (if not empty), or, for/? = 1, a compact infinite subset of A, are sets of uniqueness. We shall see another example in Section 9.9 for K = C. The preceding result shows that if an open connected subset A c= Cp is such that A n Rp ^ 0, any analytic function / in A is completely deter- mined by its values in A n W. The restriction of/to A n Rp is an analytic function, but in general an analytic function in A n W cannot be extended to an analytic function in A; we have however the weaker result: (9.4.5) Let Ebea complex Banach space, A an open subset of W, fan analytic mapping of A into E. Then there is an open set B c Cp such that B n Rp = A, and an analytic mapping gofE into E which extends f. Indeed, for each a = (ai9..., ap) e A, there is an open polydisk Pa in W defined by \xi — ai\